List decoding of Convolutional Codes over integer residue rings

TL;DR

This paper investigates list decoding of convolutional codes over integer residue rings over erasure channels, proposing a recursive method leveraging matrix polynomial representations and Toeplitz structures to recover codewords.

Contribution

It introduces a novel list decoding approach for convolutional codes over residue rings using $p$-adic expansion and matrix polynomial techniques.

Findings

Effective list decoding method for convolutional codes over residue rings.

Utilizes Toeplitz structure for step-by-step decoding.

Recovers codewords from erasure-affected transmissions.

Abstract

A convolutional code $\C$ over $\ZZ[D]$ is a $\ZZ[D]$-submodule of $\ZZN[D]$ where $\ZZ[D]$ stands for the ring of polynomials with coefficients in $\ZZ$. In this paper, we study the list decoding problem of these codes when the transmission is performed over an erasure channel, that is, we study how much information one can recover from a codeword $w\in \C$ when some of its coefficients have been erased. We do that using the $p$-adic expansion of $w$ and particular representations of the parity-check polynomial matrix of the code. From these matrix polynomial representations we recursively select certain equations that $w$ must satisfy and have only coefficients in the field $p^{r-1}\ZZ$. We exploit the natural block Toeplitz structure of the sliding parity-check matrix to derive a step by step methodology to obtain a list of possible codewords for a given corrupted codeword $w$, that is, a list with the closest codewords to $w$.